1. Field of the Invention
The present invention relates to a process simulator and process simulating method that predicts an internal physical quantity and configuration of an impurity profile or the like in the process of manufacturing a semiconductor device.
2. Description of the Related Art
When manufacturing a semiconductor transistor (referred to as a semiconductor device, hereinafter), the manufacturing processes of the semiconductor device including oxidation process, diffusion process, ion implantation process, and the like are simulated by a process simulator using a computer, and an internal physical quantity and configuration of an impurity profile of the semiconductor device are predicted. For example, when oxidation and diffusion are applied to an ion-implanted initial configuration, it is possible that changes in the configuration owing to the oxidation and the diffusion of the impurity in its oxidation atmosphere in each oxidation/diffusion time are solved by turns to predict the final configuration of the device and the changes in the impurity profile according to the time passing.
The use of a process simulator, for the purpose of optimizing the semiconductor device so that the device displays the best electrical characteristics, can save the trouble of actually manufacturing the LSI (large-scale integration) and verifying its electrical characteristics, thereby reducing the production cost of the semiconductor device and shortening the time required for the production. Since the process simulator calculates the internal physical quantity inside the semiconductor device, it is also possible to analyze behavior inside the semiconductor device.
In the process simulator, in order to obtain the value of a physical quantity inside the semiconductor device, it is necessary to solve partial differential equations such as diffusion equation of continuity which expresses the behavior of the impurity. However, since the partial differential equations cannot be analytically solved, the calculation is performed by dividing the semiconductor device into smaller regions and making the partial differential equations discrete. As for the technique of this kind, for example, there is a method described from pp. 51 to pp. 62 ("3. Process Simulator, Chapter 3 Process Simulation") of "VLSI Design-Manufacture Simulation" (edited by Michitada Morisue, published by CMC Ltd., 1987). In the literature, a method for calculating a one dimensional profile due to oxidation and diffusion is described.
Also, as for anther technique of the conventional art, there is a technique described from pp. 91 to pp. 122 ("3. Device Simulation") of "Process Device Simulation Techniques" (written and edited by Ryo Dan, published by Sangyo Tosho, 1988). In the literature, as an example of analyzing a two dimensional structure, a method of calculating by dividing the semiconductor device into smaller rectangular regions and making the partial differential equations discrete is described.
On the other hand, in the case of analyzing the semiconductor device having complex configuration like LOCOS (Local Oxidation of Silicon) configuration or trench configuration, there is a method of using a triangle to divide the configuration into smaller regions and making it discrete in order to correctly express the configuration of the semiconductor. As for the method, the detail is found in "Iterative Methods in Semiconductor Device Simulation" (C. S. Rafferty, M. R. Pinto, IEEE Transaction on Electron Devices, Vol. ED-32, No. 10, October, 1985). According to the literature, when simulating the trench structure, the configuration of the semiconductor can be expressed as a set of triangular elements in the case where the configuration is divided into smaller regions, e.g. discrete triangles, as shown in FIGS. 13 and 14. Therefore, the trench structure can be correctly expressed. Here, FIG. 13 is a schematic sectional view of the semiconductor device. FIG. 14 shows the state where the sectional view of the semiconductor device in FIG. 13 is expressed by a set of discrete triangular elements.
FIG. 15 is a schematic view showing a part of the semiconductor device as shown in FIGS. 13 and 14. Hereinafter, referring to FIG. 15, the solution of the partial differential equation by the finite difference method using the triangular elements will be explained.
First, an impurity concentration and an electric potential caused by the activated impurity are defined on each mesh node (a vertex of a triangle). The impurity is diffused owing to the concentration gradient and the electric potential gradient, and a flux of the impurity during diffusion is defined on a edge of the triangle.
Now, according to the Gauss' theorem, in the definition of a closed curved surface, the total amount resulted from integrating the impurity by volume in the closed curved surface is equal to the amount resulted from integrating the flux vertically crossing the closed curved surface by area. Therefore, it is necessary to define the closed curved surface perpendicular to the flux in order to apply the Gauss' theorem to the discrete way by the triangles as mentioned above. Namely, it is necessary to define the closed curved surface in the Gauss' theorem as a region surrounding the perpendicular bisectors of the sides of the triangles whose sides are respectively connected to the vertices, i.e., as a region formed by connecting the circuncenters of the respective triangles. Here, the closed curved surface of each joint (each vertex of the triangles) is generally called a control volume. In this case, the total amount of the impurity each joint governs is obtained by multiplying the impurity concentration of the joint by the volume of the control volume (if in the case of two dimension, assume that the depth is 1). Then, by calculating the total amount of the impurity one by one and adding the results with respect to all of the vertices in the system being analyzed, the total amount in the system is equal to the total dose amount in the ion implantation.
Now, the definition of an appropriate control volume (the open curved surface at each joint) necessitates a condition that the circumcenters of the adjacent triangles do not intersect each other essentially. The reason is that if the circumcenters of the adjacent triangles intersect each other, the product of the section when the flux is integrated by area becomes negative. If the condition is not satisfied, there causes such a projection of the electric potential that could not physically occur, as shown in FIG. 16. In FIG. 16, the triangular mesh is partially omitted. While, if the control volumes intersect each other, the total amount of the impurity is not equal to the total dose amount in the ion implantation, even if the total amount is obtained by multiplying the impurity concentration of the node by the volume of the control volume, calculating in this way with respect to all of the vertices in the system to be analyzed, and adding the results.
In order to avoid the above problems, it is required to guarantee the Delaunay partitioning that there is no vertex of other triangles within the circumscribed circle of the triangle at issue, and to perform the triangular division, thereby to satisfy the condition that the circumcenters of the adjacent triangles do not intersect each other. As for the method for guaranteeing the Delaunay partitioning and performing the triangular division, there is a description in "Tetrahedral elements and the Scharfetter-Gummel method" (M. S. Mock, Proceeding of the NASECODE IV, pp. 36 to pp. 47, 1985) that the new joints required for improving the calculation accuracy, i.e., the material boundary points are added one by one to a group of the triangles that have already been divided by the Delaunay partitioning.
In other words, as shown in FIG. 17(A), a new joint is added to a group of the triangles that have already been divided by the Delaunay partitioning, and triangles having circumscribed circles including the new joint are searched. Next, as shown in FIG. 17(B), the searched triangles are deleted, and the outermost contour of the region formed by the deleted triangles is extracted (the portion indicated by the bold lines in the drawing). Then, as shown in FIG. 17(C), new triangles are formed by connecting the sides of the extracted outermost contour with the new joint. The group of the triangles newly formed according to the above method are also divided by the Delaunay partitioning.
However, according to the conventional method of the process simulation mentioned above, since the configuration of the semiconductor device varies in calculating the oxidation process, the discrete configuration by a triangular varies as well. Even if the Delaunay partitioning is guaranteed to the configuration of the semiconductor device before oxidation, the triangular configuration varies according to the oxidation, and therefore, the Delaunay partitioning can't be guaranteed after oxidation. When solving the diffusion equation after oxidation, it is necessary to perform the triangular division once more so as to guarantee the Delaunay partitioning to the configuration of the deformed semiconductor device.
Further, even if executing the triangular division with the Delaunay partitioning guaranteed to the deformed configuration after oxidation, the reformed configurations of the new triangular elements are different from those before oxidation, so that the control volumes vary according to the change in the configuration. Therefore, even if multiplying the impurity concentration of each joint by the volume of the control volume one by one and adding these values after oxidation with respect to all of the vertices in the system to be analyzed, the total dose amount of the impurity before the deformation owing to the oxidation cannot be reserved.
In addition, in order to improve the analytic accuracy in solving the diffusion equation, a new joint may need to be added when reforming the triangles on the deformed semiconductor device after oxidation. In this case, it is necessary to define the impurity concentration of the added joint, which takes a lot of trouble defectively.